Recent zbMATH articles in MSC 17Ahttps://www.zbmath.org/atom/cc/17A2021-04-16T16:22:00+00:00WerkzeugBipartite graphs and the structure of finite-dimensional semisimple Leibniz algebras.https://www.zbmath.org/1456.170052021-04-16T16:22:00+00:00"Turdibaev, Rustam"https://www.zbmath.org/authors/?q=ai:turdibaev.rustam-mThis paper is devoted to structure of a finite-dimensional semisimple Leibniz algebras and their construction using the bipartition of an associated graph. A Leibniz algebra is a nonantisymmetric generalization of a Lie algebra. It is well known that semisimple Lie algebras can be written as direct sums of simple Lie algebras. However, this in not necessarily true for semisimple Leibniz algebras. Structure of semisimple Leibniz algebras is open and so the author uses a bipartite graph to build them.
Using Weyl's Theorem on Complete Reducibility a decomposition for a semisimple Leibniz algebra is given. First they prove that for an indecomposable semisimple Leibniz algebra with the given decomposition; the graph, whose vertices are simple Lie algebras in that decomposition, is connected. Then it is shown that a finite dimensional semisimple Leibniz algebra is indecomposable if and only if the associated bipartite graph is connected. Finally, they show that one can construct a finite-dimensional indecomposable semisimple Leibniz algebra via any finite connected bipartite graph and give an example showing this construction.
Reviewer: Ismail Demir (Izmir)Cup-product for equivariant Leibniz cohomology and Zinbiel algebras.https://www.zbmath.org/1456.170032021-04-16T16:22:00+00:00"Mukherjee, Goutam"https://www.zbmath.org/authors/?q=ai:mukherjee.goutam"Saha, Ripan"https://www.zbmath.org/authors/?q=ai:saha.ripanThe goal of the the article under review is the introduction of equivariant Leibniz cohomology and a Zinbiel product on it.
Recall that a (right) \textit{Leibniz algebra} is is a \(k\)-vector space \({\mathfrak g}\) with a \(k\)-bilinear bracket \([,]\) such that for all \(x,y,z\in{\mathfrak g}\)
\[[x,[y,z]]=[[x,y],z]-[x,z],y].\]
Jean-Louis Loday introduced Leibniz algebras to study the failure of periodicity in algebraic K-theory. He noticed that the Chevalley-Eilenberg coboundary operator lifts to tensor powers (when putting the bracket in the \(i\)th place) to give the coboundary operator \(d:{\mathfrak g}^{\otimes n}\to {\mathfrak g}^{\otimes(n-1)}\) for the homology of Leibniz algebras
The authors consider linear actions of a finite group \(G\) by automorphisms on a Leibniz algebra \({\mathfrak g}\). They develop in detail a differential geometric example (where they naturally switch to \textit{left} Leibniz algebras), namely the Leibniz algebroid structure on \(\Lambda^{n-1}T^*M\) for a Nambu-Poisson manifold \(M\) of order \(n\) with a smooth action of a finite group \(G\).
Then follows the definition of the equivariant cohomology. For this, the authors use Bredon's equivariant cohomology set-up. Namely, to \(G\), one associates a category \(O_G\) whose objects are the left cosets \(G/H\) for \(H\) running through all subgroups \(H\) of \(G\), and morphisms \(G/H\to G/K\) being \(G\)-maps (for the \(G\)-action on \(G/H\) given by left-translation). An \(O_G\)-module is then a contravariant functor \(O_G\to k\)-mod. In the symmetric monoidal category of \(O_G\)-modules, one can consider associative commutative algebras \(A\), but also (right) Leibniz algebras. In fact, the data of a Leibniz algebra \({\mathfrak g}\) in \(k\)-modules equipped with the action of a finite group \(G\) gives rise to a Leibniz algebra in \(O_G\)-modules.
The equivariant cohomology is then defined with the help of standard complexes. On the one hand, one can transpose the setting of the Loday standard complex for Leibniz cohomology into the category of \(O_G\)-modules. On the other hand, for an associative commutative \(O_G\)-algebra \(A\), one can consider in the collection of
\[S^n({\mathfrak g},A):=\bigoplus_{H < G}CL^n({\mathfrak g},A(G/H))\]
the subcomplex of invariant cochains. The authors show in their Theorem 4.5 that both complexes compute the same cohomology, the \textit{\(G\)-equivariant Leibniz cohomology} of \({\mathfrak g}\).
The last section is then devoted to the construction to the graded Zinbiel cup product on equivariant Leibniz cohomology.
Reviewer: Friedrich Wagemann (Nantes)The Grassmann algebra over arbitrary rings and minus sign in arbitrary characteristic.https://www.zbmath.org/1456.160192021-04-16T16:22:00+00:00"Dor, Gal"https://www.zbmath.org/authors/?q=ai:dor.gal"Kanel-Belov, Alexei"https://www.zbmath.org/authors/?q=ai:kanel-belov.alexei"Vishne, Uzi"https://www.zbmath.org/authors/?q=ai:vishne.uziThe Grassmann (also known as exterior) algebra \(G\) is widely used in several areas of mathematics and also in theoretical physics. It provides the most natural and fundamental example of a superalgebra. It is usually defined as the associative algebra generated by the elements \(e_1\), \(e_2\), \dots, subject to the relations \(e_ie_j=-e_je_i\) for every \(i\) and \(j\); in particular \(e_i^2=0\). The Grassmann algebra became extremely important in PI theory. It is the easiest example of an algebra that satisfies polynomial identities (the triple commutator \([x,y,z]\)) but in characteristic 0 does not satisfy any standard identity. Later on it was an essential tool in the theory developed by \textit{A. R. Kemer} [Ideals of identities of associative algebras. Transl. from the Russian by C. W. Kohls. Transl. ed. by Ben Silver. Providence, RI: American Mathematical Society (1991; Zbl 0732.16001)]. It also appears in the definition of a superalgebra (not necessarily associative).
It is well known that the identities of \(G\) follow from the triple commutator whenever the base field is infinite and of characteristic different from 2. The problems arise in characteristic 2: the above definition of \(G\) makes it commutative. In 2000, the second-named author of the paper under review constructed an analogue \(G^+\) of the Grassmann algebra in characteristic 2, and used it to construct one of the first examples of associative algebras (in characteristic 2) whose identities do not admit any finite basis, see [\textit{A. Ya. Belov}, Sb. Math. 191, No. 3, 13--24 (2000; Zbl 0960.16029); translation from Mat. Sb. 191, No. 3, 13--24 (2000)].
Generalising the Grassmann algebra \(G\) one comes naturally to the notion of the free superalgebra \(S\), it is the tensor product of a polynomial algebra and of the Grassmann algebra. Denote the supercommutator of \(a\) and \(b\) by \(\{a,b\}\).
The authors consider algebras over a commutative and associative unital ring \(C\). They construct an algebra \(\mathfrak{G}\) over \(C\) whose behaviour is similar to that of \(G\). We recall here that the above mentioned algebra \(G^+\) can take the place of \(G\) only in characteristic 2. So the algebra \(\mathfrak{G}\) is indeed a characteristic-free generalisation of \(G\). They prove that the ideal of identities of \(\mathfrak{G}\) is generated by the triple commutator. If \(1/2\in C\) they prove that \(\mathfrak{G}\) is PI equivalent to the free supercommutative algebra \(S\) and moreover for every \(C\)-algebra \(A\) the identities of \(A\otimes_C \mathfrak{G}\) and of \(A\otimes_C S\) are the same.
If \(C\) is a field of characteristic not 2, it is well known that for every permutation \(\sigma\) of the symmetric group \(S_n\), the equality \(e_{\sigma(1)} \cdots e_{\sigma(n)} = (-1)^\sigma e_1\cdots e_n\) holds in \(G\) where \((-1)^\sigma\) is the sign of the permutation \(\sigma\). The authors define an analogue to the sign, denoted by \(\mathfrak{sgn}\), in the setting of \(\mathfrak{G}\). This generalised sign admits a cohomological interpretation exploited by the authors. They use it in order to show that the codimension sequence of \(\mathfrak{G}\) behaves exactly like the one for \(G\), namely the \(n\)-th codimension of \(\mathfrak{G}\) equals \(2^{n-1}\).
The authors also consider generalised superalgebras (called in the paper \(\Sigma\)-superalgebras) in the following sense: these are graded by an infinite but enumerable group of exponent 2 (which must be abelian). They construct the free \(\Sigma\)-supercommutative algebra \(\mathfrak{S}\). Afterwards they generalise the construction of the Grassmann hull (used first by Kemer, see the above reference), to the generalised Grassmann hull of a \(\Sigma\)-superalgebra. The authors study in detail the ideal of identities of this algebra.
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In the case of algebras over a field of characteristic different from 2, one defines the supertrace in a canonical way using the properties of the Grassmann algebra. The authors develop the notion of a supertrace to \(\Sigma\)-supertraces; it is characteristic-free. They define the free \(\Sigma\)-supertrace algebra and the corresponding notion of \(\Sigma\)-supertrace identities. They also develop a theory of identities with a linear function.
Reviewer: Plamen Koshlukov (Campinas)Derivations and automorphisms of nilpotent evolution algebras with maximal nilindex.https://www.zbmath.org/1456.170192021-04-16T16:22:00+00:00"Mukhamedov, Farrukh"https://www.zbmath.org/authors/?q=ai:mukhamedov.farruh-m"Khakimov, Otabek"https://www.zbmath.org/authors/?q=ai:khakimov.otabek-n"Omirov, Bakhrom"https://www.zbmath.org/authors/?q=ai:omirov.bakhrom-a"Qaralleh, Izzat"https://www.zbmath.org/authors/?q=ai:qaralleh.izzatAn algebra \((E,+,\cdot)\) over a field \(K\) is called an \textit{evolution algebra} provided it has a basis \(\{e_i\}\) such that \(e_i\cdot e_j=\) for everi \(i\neq j\) and \(e_i\cdot e_i=\sum_k a_{i,k}e_k\). Such a basis, is called \textit{natural basis}. This paper deals with nilpotent evolution algebras \(E\) such that \(\dim E^2= \dim E -1\).
The authors describe the Lie algebra of derivations of \(E\) when \(E\) has maximal index of nilpotency. Furthermore, they describe local and 2-local derivations for such algebras showing, for example, that every 2-local derivation of \(E\) is a derivation.
Finally, the authors determine the automorphisms and local automorphims for this type of algebras.
Reviewer: Antonio M. Oller Marcén (Zaragoza)Cohomologically rigid solvable Leibniz algebras with nilradical of arbitrary characteristic sequence.https://www.zbmath.org/1456.170042021-04-16T16:22:00+00:00"Mamadaliev, U. Kh."https://www.zbmath.org/authors/?q=ai:mamadaliev.u-kh"Omirov, B. A."https://www.zbmath.org/authors/?q=ai:omirov.bakhrom-aThe authors describe the \((n + s)\)-dimensional solvable Leibniz algebras whose nilradical has characteristic sequence \((m_1,\ldots, m_s)\), where \(m_1+\ldots+ m_s=n\). Moreover, the authors prove that such Leibniz algebras \(L\) have the trivial center, every derivation of \(L\) is inner, and the second cohomology group \(HL^2(L,L)\) is trivial.
Reviewer: Alexandre P. Pojidaev (Novosibirsk)Color Lie rings and PBW deformations of skew group algebras.https://www.zbmath.org/1456.170162021-04-16T16:22:00+00:00"Fryer, S."https://www.zbmath.org/authors/?q=ai:fryer.sian"Kanstrup, T."https://www.zbmath.org/authors/?q=ai:kanstrup.tina"Kirkman, E."https://www.zbmath.org/authors/?q=ai:kirkman.ellen-e"Shepler, A. V."https://www.zbmath.org/authors/?q=ai:shepler.anne-v"Witherspoon, S."https://www.zbmath.org/authors/?q=ai:witherspoon.sarah-jThe authors study color Lie rings over finite group algebras and the corresponding universal enveloping algebras [\textit{M. Scheunert}, J. Math. Phys. 20, 712--720 (1979; Zbl 0423.17003)]. More precisely, they consider such rings that arise from finite abelian groups acting diagonally on a finite dimensional vector space over a field of characteristic 0 and prove that their universal enveloping algebras can be presented as quantum Drinfeld orbifold algebras [\textit{P. Shroff}, Commun. Algebra 43, No. 4, 1563--1570 (2015; Zbl 1332.16020)]. The proof is mainly based on the theory of PBW deformations and related tools (see, e.g. [\textit{P. Shroff} and \textit{S. Witherspoon}, J. Algebra Appl. 15, No. 3, Article ID 1650049, 15 p. (2016; Zbl 1345.16025)]. As an application they show that these algebras are braided Hopf algebras.
Reviewer: Aleksandr G. Aleksandrov (Moskva)Some remarks about the levels and sublevels of algebras obtained by the Cayley-Dickson process.https://www.zbmath.org/1456.170062021-04-16T16:22:00+00:00"Flaut, Cristina"https://www.zbmath.org/authors/?q=ai:flaut.cristinaLet \(A\) be an algebra over a field \(K\) of characteristic not \(2\). The level \(s\) (resp. sublevel \(\underline{s}\)) of \(A\) is defined to be the smallest \(n\) such that \(-1\) (resp. \(0\)) is a sum of \(n\) squares (resp. \(n+1\) nonzero squares) in \(A\), or \(\infty\) if no such \(n\) exists. So in particular \(\underline{s}(A)\leq s(A)\). In this paper, the author continues her study of the possible values of the level and sublevel for algebras of dimension \(2^t\) (\(t\geq 2\)) obtained by the Cayley-Dickson doubling process. Examples of such algebras are quaternion algebras (\(t=2\)) and octonion algebras (\(t=3\)). For example, it is shown that for any positive integers \(n\geq 1\), \(t\geq 2\) with \(n\leq 2^t\), there is a division algebra \(A\) of dimenson \(2^t\) obtained by the Cayley-Dickson doubling process such that \(s(A)=\underline{s}(A)=n\). It is also shown that such algebras always exist for \(n>2^t\) when \(n\) is a power of \(2\), but it is still an open problem for values \(n>2^t\) that are not powers of \(2\). Some of the results are fairly straightforward generalizations of results about levels and sublevels of octonions algebras due to \textit{J. O'Shea} [Math. Proc. R. Ir. Acad. 110A, No. 1, 21--30 (2010; Zbl 1276.17001)] or \textit{S. Pumplün} [Proc. Am. Math. Soc. 133, No. 11, 3143--3152 (2005; Zbl 1134.17302)] using similar techniques.
Reviewer: Detlev Hoffmann (Dortmund)On free Gelfand-Dorfman-Novikov-Poisson algebras and a PBW theorem.https://www.zbmath.org/1456.170012021-04-16T16:22:00+00:00"Bokut, L. A."https://www.zbmath.org/authors/?q=ai:bokut.leonid-a"Chen, Yuqun"https://www.zbmath.org/authors/?q=ai:chen.yuqun"Zhang, Zerui"https://www.zbmath.org/authors/?q=ai:zhang.zeruiGelfand-Dorfman-Novikov algebras (GDN algebras, known also as Novikov algebras) were introduced independently in [\textit{I. M. Gel'fand} and \textit{I. Ya. Dorfman}, Funkts. Anal. Prilozh. 13, No. 4, 13--30 (1979; Zbl 0428.58009); translation in Funct. Anal. Appl. 13, 248--262 (1980; Zbl 0437.58009)]
in connection with Hamiltonian operators in the formal calculus of variations and in [\textit{A. A. Balinskii} and \textit{S. P. Novikov}, Sov. Math., Dokl. 32, 228--231 (1985; Zbl 0606.58018); translation from Dokl. Akad. Nauk SSSR 283, 1036--1039 (1985)] in connection with linear Poisson brackets of hydrodynamic type.
These algebras satisfy the identities of left symmetry
\[ x\circ(y\circ z)-(x\circ y)\circ z = y\circ (x\circ z)-(y\circ x)\circ z\]
and right commutativity
\[ (x\circ y)\circ z = (x\circ z)\circ y. \]
GDN-Poisson algebras were introduced in [\textit{X. Xu}, J. Algebra 185, No. 3, 905--934 (1996; Zbl 0863.17003); J. Algebra 190, No. 2, 253--279 (1997; Zbl 0872.17030)]. These are GDN algebras with an additional operation \(\cdot\) which equips the algebra with the structure of a commutative associative algebra with the compatibility conditions
\[ (x\cdot y)\circ z = x\cdot (y\circ z)\quad\text{and}\quad
(x\circ y)\cdot z-x\circ (y\cdot z) = (y\circ x)\cdot z-y\circ (x\cdot z). \]
In the paper under review the authors construct a linear basis of the free GDN-Poisson algebra.
Then they define the notion of a special GDN-Poisson admissible algebra. This is a differential algebra with two commutative associative products and some extra identities.The authors prove that any GDN-Poisson algebra can be embedded into its universal enveloping special GDN-Poisson admissible algebra.
Finally, they establish that any GDN-Poisson algebra with the identity
\[ x\circ(y\cdot z)=(x\circ y)\cdot z+(x\circ z)\cdot y\]
is isomorphic to a commutative associative differential algebra both as GDN-Poisson algebra and as a commutative associative differential algebra.
Reviewer: Vesselin Drensky (Sofia)Shuffle quadri-algebras and concatenation.https://www.zbmath.org/1456.170022021-04-16T16:22:00+00:00"Mohamed, Mohamed Belhaj"https://www.zbmath.org/authors/?q=ai:belhaj-mohamed.mohamed"Manchon, Dominique"https://www.zbmath.org/authors/?q=ai:manchon.dominiqueA quadri-algebra has four products (northwest, northeast, southeast, southwest), satisfying nine axioms,
such that the sum of these four products is associative. It is also possible to sum these products two by two,
and this gives two dendriform structures on it: north-south, and east-west. A classical example of quadri-algebra is given by the shuffle Hopf algebra, where the shuffle products is cut into four according to the first and the last letter of the words.
This paper studies the relation between these four products and the concatenation product. Formulas are given for the action of the four quadri products on a concatenation of words, which imply by summation formulas for the action of the dendriform products on a concatenation and finally for the action of the shuffle product on a concatenation.
It is also proved that the south and the east products define two module-algebras over the shuffle algebra
on its augmentation ideal.
For the entire collection see [Zbl 1437.05003].
Reviewer: Loïc Foissy (Calais)Counting finite-dimensional algebras over finite field.https://www.zbmath.org/1456.170072021-04-16T16:22:00+00:00"Verhulst, Nikolaas D."https://www.zbmath.org/authors/?q=ai:verhulst.nikolaas-dAuthor's abstract: In this paper, we describe an elementary method for counting the number of non-isomorphic algebras of a fixed, finite dimension over a given finite field. We show how this method works in the case of 2-dimensional algebras over the two-element field.
Reviewer: Steven C. Althoen (Dexter)On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3.https://www.zbmath.org/1456.112242021-04-16T16:22:00+00:00"Hu, Yong"https://www.zbmath.org/authors/?q=ai:hu.yong"Wu, Zhengyao"https://www.zbmath.org/authors/?q=ai:wu.zhengyaoSummary: Let \(F\) be a field, \(\ell\) a prime and \(D\) a central division \(F\)-algebra of \(\ell\)-power degree. By the Rost kernel of \(D\) we mean the subgroup of \(F^*\) consisting of elements \(\lambda\) such that the cohomology class \((D)\cup (\lambda)\in H^3(F,\mathbb{Q}_{\ell}/\mathbb Z_{\ell}(2))\) vanishes. In ~1985, \textit{A. A. Suslin} [in: Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 25, 115--207 (1984; Zbl 0558.12013)] conjectured that the Rost kernel is generated by \(i\)-th powers of reduced norms from \(D^{\otimes i}\) for all \(i\ge 1\). Despite known counterexamples, we prove some new special cases of Suslin's conjecture. We assume \(F\) is a henselian discrete valuation field with residue field \(k\) of characteristic different from \(\ell\). When \(D\) has period \(\ell \), we show that Suslin's conjecture holds if either \(k\) is a \(2\)-local field or the cohomological \(\ell\)-dimension \(\operatorname{cd}_{\ell}(k)\) of \(k\) is \(\le 2\). When the period is arbitrary, we prove the same result when \(k\) itself is a henselian discrete valuation field with \(\operatorname{cd}_{\ell}(k)\le 2\). In the case \(\ell=\operatorname{char}(k)\), an analog is obtained for tamely ramified algebras. We conjecture that Suslin's conjecture holds for all fields of cohomological dimension 3.Triple Lie systems associated with \((-1, 1)\) algebras.https://www.zbmath.org/1456.170182021-04-16T16:22:00+00:00"Borisova, L. R."https://www.zbmath.org/authors/?q=ai:borisova.ludmilla-r"Pchelintsev, S. V."https://www.zbmath.org/authors/?q=ai:pchelintsev.sergei-valentinovichIf \(J\) is a Jordan algebra one can construct a Lie triple system \(T(J)\) be defining on \(J\) the triple product
\([x,y,z]:=(xy)z-x(yz)\). For any right alternative algebra \(A\) one can construct \(T_J(A)=T(A^+)\) (where \(A^+\) is the symmetrization algebra). Recall that a \((- 1, 1)\) ring is a nonassociative ring satisfying the conditions
\[ 0 = (x, y, z) + (y, z, x) + (z, x, y),\quad 0 = (x, y, z) + (x, z, y) \]
for arbitrary elements in the ring. In a work published by \textit{S. V. Pchelintsev} and \textit{I. P. Shestakov} [J. Algebra 423, 54--86 (2015; Zbl 1369.17029)], the relation between \((-1,1)\)-algebras and Jordan algebras are studied. In the same year, a work by V. Zhelyabin and A. S. Zakharov explores the relations between \((-1,1)\)-algebras and Novikov algebras.
The present work deals with right alternative \((-1,1)\)-algebras. It studies relations between \((-1,1)\)-algebras and certain Lie triple systems that are called \(TL_K(A)\), related to central isotopes of \((-1,1)\)-algebras. It is proved that a \((-1,1)\)-algebras \(A\) has nilpotent associator ideal if and only if the Lie triple system \(TL_K(A)\) is nilpotent. The question on the solvability and nilpotency of the Lie triple systems \(T_J(A)\) and \(TL_K(A)\) constructed from \(A\) is also studied.
Reviewer: Candido Martín González (Málaga)